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Prime number theorem - Wikipedia
The prime number theorem then states that x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1: known as the asymptotic law of distribution of prime numbers.

Tighter bounds in the prime number theorem - johndcook.com
New tighter bounds for the prime number theorem, valid for all x and not just enormous values.

big list - Bounds for $n$-th prime - Mathematics Stack Exchange
If you want even tighter bounds, it turns out that the published bounds for prime counts are tighter than the nth prime bounds. By using an inverse lookup (binary search) of the opposite prime count bound, we can achieve tighter bounds for a pretty large range.

Chebychev Bounds for Prime Counting Function
The celebrated prime number theorem is the following result: Theorem (Prime Number Theorem): The number of primes bounded by a real value x is assympotitcally equivalent to x / log x.

Bounded gaps between primes - Annals of Mathematics
Our method is a re nement of the recent work of Goldston, Pintz and Y ld r m on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

Prime Number Gaps Explained
Delve into the fascinating world of prime numbers, exploring the gaps between them and the bounds that govern their distribution, a crucial aspect of number theory.

Upper Bounds for Prime Numbers - ProofWiki
Upper Bounds for Prime Numbers Contents 1 Theorem 1.1 Result 1 1.2 Result 2 1.3 Result 3 2 Motivataion

The Prime Number Theorem Ben Green - University of Oxford
7.4. Bounds for 0= at judicuously chosen points Chapter 8. he prime number theor Chapter 9. The .1. the is and its Chapter 10.

The Prime Number Theorem - A PRIMES Exposition
Prime Number Theorem with Error Term The more we increase the bounds on the zero-free region, the better our precision of our estimate will be.

Prime Number Theorem - from Wolfram MathWorld
The prime number theorem gives an asymptotic form for the prime counting function pi (n), which counts the number of primes less than some integer n.

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